## Mechanisms of Wetting Transitions Statics 651 Composite Wetting State

As mentioned, the Cassie air trapping wetting state corresponds to the highest of multiple minima of Gibbs energy of the droplet deposited on the rough surface (with biological and hierarchical surfaces being exceptions). Thus, for the WT the energy barrier should be surmounted [3, 50, 79]. It was supposed that this energy barrier corresponds to the surface energy variation between the Cassie state and the hypothetical composite state with the almost complete filling of surface asperities by water keeping the liquid-air interface under the droplet and the contact angle constant, as shown in Fig. 6.2d. Contrastingly to the equilibrium mixed wetting state [67], the composite state is unstable for hydrophobic surfaces and corresponds to energy maximum (transition state). For the simple topography depicted in Fig. 6.5, the energy barrier could be calculated as follows [26]:

Wtrans = 2pR2h(ySL - ySA)/p = —2pR2hy cos dy/p (6.10)

where h and p are the geometric parameters of the relief, shown in Fig. 6.5, and R is the radius of the contact area. The numerical estimation of the energetic barrier according to formula (6.10) with the parameters p = h = 20 im, R = 1 mm, hy = 105^corresponding to low density polyethylene (LDPE)), and

Fig. 6.5 Geometric parameters of the model relief used for the calculation of Cassie-Wenzel transition energetic barrier

Fig. 6.5 Geometric parameters of the model relief used for the calculation of Cassie-Wenzel transition energetic barrier

r4irLrLrLr

a value of Wtrans = 120 nJ.

It should be

stressed that y = 72 mJ m_~ gives according to (6.10) the energy barrier scales as Wtrans * R2. The validity of this assumption will be discussed below. The energetic barrier is extremely large compared to thermal fluctuations: Wkp ~ (R)1, where a is an atomic scale [50]. At the same time Wtrans is much less than the energy of evaporation of the droplet Q « (4/3)pR3k, where k is the volumetric heat of water evaporation, k = 2 x 109 J/m3. For a 3 il droplet with the radius R & 1 mm it yields Q « 10 J, hence kT ^ Wtrans ^ Q. Actually, this interrelation between characteristic energies makes wetting transitions possible. If it was not the case, a droplet exposed to external stimuli might evaporate before wetting transition. It is instructive to estimate the radius, at which Wtrans « Q. Equating Wtrans given by Eq. 6.10 to Q yields R « — (3/2)y cos hY/k « 5 x 10—1!m. It means that wetting transitions are possible for any volume of a droplet. It is noteworthy that the ratio y/k is practically the same for all liquids, and it is of the order of magnitude of molecular size [27]. Hence, wetting transitions are possible for any liquid and any volume.

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